In this section, we will use manipulatives to model addition, subtraction, multiplication, and division with positive and negative integers. By working with these models, we will see why the laws for signed numbers (positive and negative) work the way they do. Note 9

We'll begin with models for addition in which we will represent positive numbers with black chips and negative numbers with red chips:

Using red and black chips, every number can be represented in a variety of ways. For example, here are some ways to represent positive two (+2):

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Two black chips:

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Three black chips and one red chip:

In this case, one black chip and one red chip together sum to zero: +1 + (-1) = 0. What remains are two black chips, or +2.

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Five black chips and three red chips:

And so on! Now let's explore some addition problems using these chips.

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To find the sum of +3 + (+1), join three black chips (three positives) with one black chip (one positive), for a total, or sum, of four black chips (four positives, or +4):

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To find the sum of -5 + (-2), join five red chips (five negatives) with two red chips (two negatives), for a sum of seven red chips (seven negatives, or -7):

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To find the sum of -4 + (+6), join four red chips (four negatives) with six black chips (six positives):

Because one black chip and one red chip together sum to zero, four pairs of black and red chips zero out. What remains are two black chips (two positives, or +2):

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To find the sum of +2 + (-5), join two black chips (two positives) with five red chips (five negatives):

Because one black chip and one red chip together sum to zero, two pairs of black and red chips zero out. What remains are three red chips (three negatives). The sum is negative three (-3):

Draw or make a diagram to show the colored-chip model for each of the following: